Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra

We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.Comment: 12 pages, reference added, minor corrections (typos, notation changes, etc

Quantum Algebraic Approach to Refined Topological Vertex

We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi^* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi^*. The spectral parameters attached to Fock spaces play the role of the K"ahler parameters.Comment: 27 page

Heisenberg realization for U_q(sln) on the flag manifold

We give the Heisenberg realization for the quantum algebra $U_q(sl_n)$, which is written by the $q$-difference operator on the flag manifold. We construct it from the action of $U_q(sl_n)$ on the $q$-symmetric algebra $A_q(Mat_n)$ by the Borel-Weil like approach. Our realization is applicable to the construction of the free field realization for the $U_q(\widehat{sl_n})$ [AOS].Comment: 10 pages, YITP/K-1016, plain TEX (some mistakes corrected and a reference added

A new class of Matrix Models arising from the W-infinity Algebra

We present a new class of hermitian one-matrix models originated in the W-infinity algebra: more precisely, the polynomials defining the W-infinity generators in their fermionic bilinear form are shown to expand the orthogonal basis of a class of random hermitian matrix models. The corresponding potentials are given, and the thermodynamic limit interpreted in terms of a simple plasma picture. The new matrix models can be successfully applied to the full bosonization of interesting one-dimensional systems, including all the perturbative orders in the inverse size of the system. As a simple application, we present the all-order bosonization of the free fermionic field on the one-dimensional lattice.Comment: 8 pages, 1 figur

Free Boson Realization of Uq(slN^)U_q(\widehat{sl_N})

We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\!\rightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space.Comment: 24 pages, LaTeX, RIMS-924, YITP/K-101

Mass Deformation of the Multiple M2 Branes Theory

Based on recent developments, in this letter we study the one parameter deformation of 2+1 dimensional gauge theories with scale invariance and N = 8 supersymmetry, which is expected to be the field theory living on a stack of M2 branes. The deformed gauge theory is defined by a Lagrangian and is based on an infinite set of novel 3-algebras constructed by relaxing the assumption that the invariant metric is positive definite. Under the Higgs mechanism, we can obtain the D-branes world volume theory in the presence of background fluxes.Comment: 13pages, no figures, reference adde

Surface Operator, Bubbling Calabi-Yau and AGT Relation

Surface operators in N=2 four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.Comment: 36 pages, 14 figures. v2: minor changes and typos correcte

Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants

Inspired by the work of Alday, Gaiotto and Tachikawa (AGT), we saw the revival of Poincar{\'{e}}'s uniformization problem and Fuchsian equations obtained thereof. Three distinguished aspects are possessed by Fuchsian equations. First, they are available via imposing a classical Liouville limit on level-two null-vector conditions. Second, they fall into some A_1-type integrable systems. Third, the stress-tensor present there (in terms of the Q-form) manifests itself as a kind of one-dimensional "curve". Thereby, a contact with the recently proposed Nekrasov-Shatashvili limit was soon made on the one hand, whilst the seemingly mysterious derivation of Seiberg-Witten prepotentials from integrable models become resolved on the other hand. Moreover, AGT conjecture can just be regarded as a quantum version of the previous Poincar{\'{e}}'s approach. Equipped with these observations, we examined relations between spheric and toric (classical) conformal blocks via Calogero-Moser/Heun duality. Besides, as Sutherland model is also obtainable from Calogero-Moser by pinching tori at one point, we tried to understand its eigenstates from the viewpoint of toric diagrams with possibly many surface operators (toric branes) inserted. A picture called "bubbling pants" then emerged and reproduced well-known results of the non-critical self-dual c=1 string theory under a "blown-down" limit.Comment: 17 pages, 4 figures; v2: corrections and references added; v3: Section 2.4.1 newly added thanks to JHEP referee advice. That classical four-point spheric conformal blocks reproducing known SW prepotentials is demonstrated via more examples, to appear in JHEP; v4: TexStyle changed onl

The Integrals of Motion for the Deformed W-Algebra Wqt(slN)W_{qt}(sl_N^) II: Proof of the commutation relations

We explicitly construct two classes of infinitly many commutative operators in terms of the deformed W-algebra $W_{qt}(sl_N^)$, and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal one, since they can be regarded as elliptic deformation of local and nonlocal integrals of motion for the $W_N$ algebra.Comment: Dedicated to Professor Tetsuji Miwa on the occasion on the 60th birthda

Counting Exceptional Instantons

We show how to obtain the instanton partition function of N=2 SYM with exceptional gauge group EFG using blow-up recursion relations derived by Nakajima and Yoshioka. We compute the two instanton contribution and match it with the recent proposal for the superconformal index of rank 2 SCFTs with E6, E7 global symmetry.Comment: 16 pages, references adde